RULE_INIT - Initialize on standby HA peer

As JRahm mentioned this couldn't be the issue but as you use irule when there is a failover the new active device will select another source ip address from the SNAT pool and this will break the previous connection. You have to use connection mirroring for this to not happen but the normal SNAT will utlize more memory:

https://support.f5.com/csp/article/K84303332#s3

Just for info you can trigger scripts on standby devices when they become active by adding the scripts under /config/failover/<state> but this seems to not be the solution in this case:

https://community.f5.com/t5/technical-forum/knowledge-sharing-ways-to-trigger-and-schedule-scripts-on-the-f5/m-p/208068#M191762